Algebraic geometry
Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 483-489.

Let $X$ be one of the $28$ Atkin–Lehner quotients of a curve ${X}_{0}\left(N\right)$ such that $X$ has genus $2$ and its Jacobian variety $J$ is absolutely simple. We show that the Shafarevich–Tate group $Ш\left(J/ℚ\right)$ is trivial. This verifies the strong BSD conjecture for $J$.

Soit $X$ un des $28$ quotients d’Atkin–Lehner d’une courbe ${X}_{0}\left(N\right)$ tel que $X$ est de genre $2$ et sa jacobienne $J$ est absolument simple. On démontre que le groupe de Shafarevich–Tate $Ш\left(J/ℚ\right)$ est trivial. Ceci vérifie la conjecture BSD forte pour $J$.

Received:
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Accepted:
Published online:
DOI: 10.5802/crmath.313
Classification: 11G40,  11-04,  11G10,  11G30,  14G35
Timo Keller 1; Michael Stoll 1

1 Lehrstuhl Mathematik II (Computeralgebra), Universität Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Timo Keller; Michael Stoll. Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 483-489. doi : 10.5802/crmath.313. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.313/

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